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Mirrors > Home > MPE Home > Th. List > 9t11e99 | Structured version Visualization version GIF version |
Description: 9 times 11 equals 99. (Contributed by AV, 14-Jun-2021.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
9t11e99 | ⊢ (9 · ;11) = ;99 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9cn 12313 | . . . 4 ⊢ 9 ∈ ℂ | |
2 | 10nn0 12696 | . . . . . 6 ⊢ ;10 ∈ ℕ0 | |
3 | 2 | nn0cni 12485 | . . . . 5 ⊢ ;10 ∈ ℂ |
4 | ax-1cn 11167 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | 3, 4 | mulcli 11222 | . . . 4 ⊢ (;10 · 1) ∈ ℂ |
6 | 1, 5, 4 | adddii 11227 | . . 3 ⊢ (9 · ((;10 · 1) + 1)) = ((9 · (;10 · 1)) + (9 · 1)) |
7 | 3 | mulridi 11219 | . . . . . 6 ⊢ (;10 · 1) = ;10 |
8 | 7 | oveq2i 7415 | . . . . 5 ⊢ (9 · (;10 · 1)) = (9 · ;10) |
9 | 1, 3 | mulcomi 11223 | . . . . 5 ⊢ (9 · ;10) = (;10 · 9) |
10 | 8, 9 | eqtri 2754 | . . . 4 ⊢ (9 · (;10 · 1)) = (;10 · 9) |
11 | 1 | mulridi 11219 | . . . 4 ⊢ (9 · 1) = 9 |
12 | 10, 11 | oveq12i 7416 | . . 3 ⊢ ((9 · (;10 · 1)) + (9 · 1)) = ((;10 · 9) + 9) |
13 | 6, 12 | eqtri 2754 | . 2 ⊢ (9 · ((;10 · 1) + 1)) = ((;10 · 9) + 9) |
14 | dfdec10 12681 | . . 3 ⊢ ;11 = ((;10 · 1) + 1) | |
15 | 14 | oveq2i 7415 | . 2 ⊢ (9 · ;11) = (9 · ((;10 · 1) + 1)) |
16 | dfdec10 12681 | . 2 ⊢ ;99 = ((;10 · 9) + 9) | |
17 | 13, 15, 16 | 3eqtr4i 2764 | 1 ⊢ (9 · ;11) = ;99 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 (class class class)co 7404 0cc0 11109 1c1 11110 + caddc 11112 · cmul 11114 9c9 12275 ;cdc 12678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-ltxr 11254 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-dec 12679 |
This theorem is referenced by: 3dvds2dec 16280 1259lem3 17072 |
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